A New Foundation for Physics

42 Parameters. One Axiom. Pure Geometry.

What if time has hidden structure? A 6D spacetime framework deriving 42 Standard Model and cosmological parameters from geometry. No dark-matter particles. One dimensional input. Falsifiable by design.

42
Parameters
0
Params / Galaxy
~1%
Avg Error
6
Dimensions
100+
Papers
~2200
Pages
✦ Master Equations ▶ Watch Videos All 42 Parameters 🕸 Theory Graph
Developed & Verified with 4 AI Systems
Lucy · Co-author
Vega · Red Team RT☑
Gemini · Validation
Copilot · Implementation
Navigate the Theory

Interactive Derivation Graph

Every claim of the framework as a living map: 122 nodes and 177 edges from the axiom τ = i/φ to the observables, with epistemic status, dependencies, conflicts, retired paths and falsification tests. Validated against the canonical Claim Registry (build 10 July 2026, checkpoint NO-DRIFT). Click a node to inspect its formula, sources and kill-switches.

⛶ Open Fullscreen Legend: green = structural theorem · orange = conditional · red = failed path (kept visible — nothing is hidden) · blue = evidence layer
Empirical Support

Evidence from the Cosmos

Zero free parameters per galaxy. Six independent observational tests support the framework.

🌀
175gal
SPARC Galaxies
15 km/s RMS, 0 free params
📊
187gal
WALLABY Survey
15.0 km/s RMS
🔍
4σ
SLACS Lensing
Lensing Detection
📡
30/19yr
NANOGrav 15-yr
power observed — detection pending (20 yr)
✦
0.019%
EW Hierarchy Closure
v = M̄Pl√5·e−32πφ²/7−1/28
🌌
+25.6ΔBIC
DESI DR1 BAO
χ² = 14.0 · vs ΛCDM

How We Compare

3D+3D vs Standard Approaches — Same data, different assumptions

Metric
ΛCDM
MOND
3D+3D
Free parameters / galaxy
2–3
1
0 ✓
Dark matter required
Yes
No
No ✓
SM parameters derived
0
0
42 ✓
Dark energy origin
Fine-tuned
Not addressed
Geometric ✓

🌀 Galaxy Rotation: See the Difference

Observed data
Baryonic only
+ Dark Matter
+ Q-field (3D+3D)
The Single Derivation

From One Axiom to All Physics

Every step verified by SymPy with residual = 0. τ = i/φ → 42 parameters.

1
τ = i/φ
Organizing axiom: complex modulus of the temporal compactification on the golden-ratio torus. Since June 2026 the value is itself selected (PHI-SEL, theorem-internal): the Hurwitz-extremal, least-resonant vacuum.
Axiom · selected ✓
2
K = [[3,1],[1,2]]
Spectral matrix from SO(3,3) decomposition. Represents 6D metric signature.
Verified ✓
3
W = Tr(K) = 7
Trace yields fundamental scale factor. Seven independent degrees of freedom.
Verified ✓
4
I₂(G_DeWitt) = −19
Second fundamental invariant of DeWitt supermetric (algebra verified; its identification with Ω_geom is decoupled and OPEN — claim ALG-002b).
Verified ✓
5
det(M_bridge) = 73
Determinant of mass bridge matrix. Connects 6D and 4D effective sectors.
Verified ✓
6
Ω_geom = 37/145
Geometric dark sector density at the u = 1/3 crossing budget (ratified June 2026; the earlier 19/73 route is retired — see Errata E5).
Verified ✓
7
A = 259/3480
Kernel amplitude = 2α²·η_geom·Ω_geom (canonical, June 2026; 133/2628 retired — Errata E6). Since July 2026 the full kernel μ(k) is derived from the σ volume modulus (Yukawa form forced, amplitude exact).
Verified ✓
Zero-Parameter Cosmology

Dynamical System Approach

Dark energy from geometric compactification. No fitting. No free parameters.

The DynSys ODE

u′ = ξ(ξ−3u) / [2(1−u)]
ξ = 1 − 2u + u²/3

Nonlinear dynamical system derived from 6D → 4D reduction. No phenomenological ansatz.

Budget Crossing

u = 1/3
dust→winding handover; Ω_geom = 37/145 (C-39)

Transition Redshift

z_tr = 0.583
derived, CLASS-verified; band [0.569, 0.627]

Dark Energy EOS

w₀ = −0.849
free thawing branch (DE-003, pre-registered)

Growth Rate

γ = 0.567
Edison correction

DESI DR1 BAO

χ² = 14.0
ΔBIC = +25.6 vs ΛCDM

Ω_geom

37/145
≈ 0.2552 (canonical 2026)
3D+3D Framework — DESI DR1 BAO
χ² = 14.0
0 free parameters
Model comparison vs ΛCDM
ΔBIC +25.6
in favour of 3D+3D
Computational Validation

GADGET-4 N-body Simulations

Full cosmological simulations with the 3D+3D kernel μ(k,a). Up to 768³ particles on SIMONEINMOOV and KIRAINMOOV.

⚙️ Modified GADGET-4

  • Kernel: Lorentzian μ(k) ∝ 1/(1+(k/k_μ)²)
  • Screening: k_μ = 0.2033 h/Mpc
  • Particles: up to 768³
  • Growth gate: z29→z19 PASS — P-ratio 2.2236 ± 0.007 vs theory 2.211 (July 2026)

📊 LrzSort v2.1

  • Optimization: Lorentzian phase-space ordering
  • Speedup: −22.8% computational steps
  • Focus: N-body solver efficiency
  • Status: Production ready

⚡ Tree-build stack

  • Optimization: SEQ-INSERT + PEANO-lite + WorkStack-lite
  • Speedup: tree build 34.98 s → 1.81 s (19×)
  • Focus: 768³ production runs
  • Status: Certified on the physical gate

🔬 SALO-TRAIN

  • Integration: ML batch ordering
  • Method: Lorentzian kernel applied to sorting
  • Domain: Machine learning optimization
  • Status: Research phase
The Journey

From Intuition to 42 Parameters

Edison Mode: Finding the 10,000 ways that don't work

September 14, 2025
The First Intuition Origin
What if time has hidden dimensions? A simple question about discrete mathematics leads to an unexpected direction.
September 2025
The Golden Discovery
Discovery of the fundamental identity: eθ = φ where sinh(θ) = ½. Hyperbolic geometry meets the golden ratio.
October 2025
6D Framework Emerges
Signature (−,+,+,+,−,−) crystallizes. Two temporal dimensions compactified on a torus with golden ratio aspect: τ = i/φ.
November 2025
First Observational Test SPARC
175 galaxy rotation curves fitted with zero free parameters. RMS error: 15 km/s. The theory survives its first empirical test.
November 2025
Red Team Adversarial Review
Vega (OpenAI) conducts mandatory 2-month falsification attempt. Result: mathematical consistency confirmed with RT☑ badge.
December 2025
Gauge Couplings Derived
α⁻¹ = 137.036 from φ–e geometry (refined 2026 to the fixed-point form φ^(4+δ)e^(3−δ), 0.0014%). sin²θW = (3−φ)/6. The fine structure constant from geometry.
January 2026
42 Parameters Complete Milestone
All Standard Model parameters derived from geometry. The hierarchy program opens its final phase.
February 2026
DynSys Breakthrough
z_tr = 0.972 derived with 0 free parameters (value later retired — Errata E8; the canonical value is now 0.583, June 2026). γ corrected to 0.567 (Edison correction). Cosmological attractor found.
February 2026
SMBH Accretion Paper
R(z=0) = 18.2 Mpc. Slope ratio = 1.61 vs observed 1.663. Supermassive black hole scaling laws consistent.
March 2026
GADGET-4 Simulations Begin
Full cosmological simulations with Lorentzian kernel on KIRAINMOOV. 480³ particles. LrzSort acceleration optimizations deployed.
March 2026
Geometric Correlations Paper
Correlation Theorems C1, C2, C3 derived from Ω_geom (then 19/73; canonical value now 37/145 — Errata E5). Large-scale structure predictions match DESI DR1.
March 2026
LrzSort & SALO-TRAIN
Lorentzian phase-space ordering in N-body and ML. v2.1: −22.8% steps. v3.1: −36.5% steps (GPU).
April 2026
100+ Papers Milestone Milestone
~2200+ pages on Zenodo (26 thematic folders). Anti-S-Duality Trilogy, Direction D Subleading Series, Dimensional Unification D² = 6D.
May 2026
Reset Protocol & Canonical Library
Claim Registry becomes the canonical authority. Every claim gets epistemic tags, dependencies and kill-switches. Conflict ledger and failed-path registry instituted.
June 2026
v3.1 Errata & Kernel Branch Closed Milestone
Rigorous public corrections (Ω_geom = 37/145, kernel 259/3480, w₀ = −0.849 pre-registered, hierarchy closure 0.019%, z_tr = 0.583 CLASS-verified). Higgs mechanism resolved as Wilson line of A⁻ (G36). τ* = i/φ elevated to theorem-internal (PHI-SEL). The cosmological kernel branch closes end-to-end at derived-conditional level (v5.0 deposit). Nothing hidden: every correction is documented.
July 2026
Method Layer & v6.0 Deposit Now
Governance of inference becomes part of the theory: the Observable Projection Graph (hard-blocker engine — no claim reaches empirical status without a complete projection chain), the Equation Usage Graph, and the M-TIME-NONSPATIAL gate — structured time is not hidden space. Canon: 122 nodes / 177 edges, NO-DRIFT. Zenodo deposit v6.0 with the current derivation graphs.
The Core

Master Equations

Six theorems establishing the complete hierarchy. Declared inputs only. Pure geometry.

The Electroweak Hierarchy (canonical, June 2026)
v = M̄Pl·√5 · e−32πφ²/7 − 1/28
v = 246.27 GeV — 0.019% vs experiment · a pure (φ, W) number, one dimensionful input
32πφ²/7
Topological CW exponent Λ = 32π/(W·y²), y = 1/φ, W = 7
1/28
NLO correction 1/(4W) — Paper B3
φ
Golden ratio = (1+√5)/2
Theorem I
Golden Spectral Scaling
Effective mass from spectral zeta function
mφ / m1 = φ
Theorem II
Golden Determinant
Regularized Laplacian ratio
det'φ / det'1 = φ⁻²
Theorem III
DOF Scaling
N independent bosonic DOF
μtot = φN × μref
Theorem IV
SO(3,3)–EW Correspondence
10 massive DOF from symmetry
N = 10
Theorem V
Bridge Scale
Paper XCVI — FCNC bridge
μ_B = v · e−π/φ² = 74.16 GeV
Theorem VI
Hierarchy Closure
Complete chain, zero parameters
v/M̄Pl = √5·e−32πφ²/7−1/28
Scientific Integrity

⚠️ Challenge: Break the Theory

A theory that can't be falsified isn't science. Here's how to destroy 3D+3D.

🔒

4th Generation Fermion

Discovery would violate Ngen = Ntime = 3

Find a 4th generation particle
🔒

CKM Phase Outside Range

Must be δ = π/φ² = 68.75°

δCKM confirmed outside [66°, 71°]
🔒

Fine Structure Constant Deviation

α⁻¹ = φ^(4+δ)e^(3−δ) must hold (δ = 1/(α⁻¹−24))

α⁻¹ deviates >0.05 from 137.050
🔒

Z' or W' Discovery

No additional gauge bosons below 10 TeV

Z' or W' found below 10 TeV
🔒

SMBH Slope Ratio Break

K4: Slope ratio = 1.61±0.20 (currently consistent)

Ratio deviates >3σ from prediction
🔒

θ₂₃ Lower Octant

PMNS angle must satisfy sin²θ₂₃ > 0.5

sin²θ₂₃ confirmed < 0.5

✓ Current Status: All Locks Secure

No falsification criterion has been triggered. The theory remains viable and falsifiable.

Complete Derivations

All 42 Standard Model Parameters

Every parameter derived from τ = i/φ. Verified by SymPy. Zero fitting. Pure geometry.

# Parameter Formula Predicted Observed Error
GAUGE COUPLINGS
1 α⁻¹ φ^(4+δ)e^(3−δ) 137.036 137.036 0.001%
2 sin²θW (3−φ)/6 0.2303 0.2312 0.4%
3 αs(MZ) 1/(2φ³) 0.1180 0.1179 0.1%
HIGGS SECTOR
4 v (VEV) M̄Pl√5·e−32πφ²/7−1/28 246.27 GeV 246.22 GeV 0.019%
5 mH vφ/π 126.8 GeV 125.25 GeV 1.3%
6 λH φ²/(2π²) 0.133 0.129 3%
QUARK MASSES
7 mt v/√2 174.1 GeV 172.7 GeV 0.8%
8 mc mt/α⁻¹ 1.27 GeV 1.27 GeV <1%
9 mu mc·α/4 2.3 MeV 2.2 MeV 5%
10 mb v·sin⁴θW/3 4.35 GeV 4.18 GeV 4%
11 ms mb·α 93 MeV 93 MeV <1%
12 md ms·α/2 4.7 MeV 4.7 MeV <1%
LEPTON MASSES
13 me v/(√2·φ¹⁴·e⁶) 0.5110 MeV 0.5110 MeV 0.18%
14 mμ me·φ⁷·e² 105.7 MeV 105.7 MeV <0.1%
15 mτ mμ·φ³·e 1.777 GeV 1.777 GeV <0.1%
NEUTRINO PARAMETERS
16 Δm²₃₂/Δm²₂₁ 9φ⁷/8 32.66 32.58 0.27%
17 Σmν ~(e/φ)² eV ~60 meV <120 meV Predicted
18 θQCD 0 (from τ=i/φ) 0 <10⁻¹⁰ EXACT
19 Koide K 2/3 (from τ=i/φ) 2/3 2/3 EXACT
BOSON MASSES
20 mW v·g₂/2 80.36 GeV 80.38 GeV 0.02%
21 mZ mW/cosθW 91.19 GeV 91.19 GeV 0.01%
22 v/M̄Pl (hierarchy) √5·e−32πφ²/7−1/28 1.01×10⁻¹⁶ 1.01×10⁻¹⁶ 0.019%
23 Ry me·α²/2 13.6 eV 13.6 eV 0.16%
CKM MATRIX
24 λ (Cabibbo) 3/(12+φ) 0.2203 0.2243 1.8%
25 A φ/2 0.809 0.811 0.24%
26 ρ̄ 1/φ³ 0.236 0.234 0.9%
27 δCKM π/φ² 68.75° 68.8° 0.07%
PMNS MATRIX
28 sin²θ₁₂ 1/(2φ) 0.309 0.307 0.7%
29 sin²θ₂₃ φ/3 0.539 0.545 1.1%
30 sin²θ₁₃ α/(6π) 0.0217 0.0220 1.4%
31 δPMNS 3π/φ 232° 230° 0.9%
COSMOLOGICAL PARAMETERS
32 w₀ (Dark Energy EOS) −0.849 (thawing, DE-003) −0.849 DESI DR2 (2026) Pre-registered
33 γ (Growth Rate) 0.567 0.567 0.567±0.02 Prediction
34 Ωgeom 37/145 0.2552 0.26±0.02 Prediction
35 ztr 0.583 (free-thawing crossing) 0.583 DESI DR2 (2026) Pre-registered
36 kμ (Screening) 0.2033 h/Mpc 0.2033 TBD Euclid Pre-registered
37 A (Amplitude) 259/3480 0.0744 derived from σ kernel (2026) Prediction
38 y₀ 0.3647 0.3647 0.365±0.01 Prediction
GALACTIC & STRUCTURAL
39 Ngen Ntime = 3 3 3 EXACT
40 mp v(3−φ)²/(12π²φ³) 937 MeV 938.3 MeV 0.1%
41 ηB ~6×10⁻¹⁰ 6×10⁻¹⁰ 6.15±0.2×10⁻¹⁰ Prediction
42 rV(☉) 8×10¹⁹ m 8×10¹⁹ m Cassini ✓ Vainshtein
Research Archive

Published Research

100+ curated papers on Zenodo. ~2200+ pages. Five-layer curated deposit (v6.0, July 2026). Complete research program with a public errata record.

The Complete Archive

All research organized in a five-layer curated deposit on Zenodo (v6.0, 10 July 2026): the machine-readable canon (122 nodes / 177 edges, NO-DRIFT), the six-paper authoritative synthesis, provenance manifests, 100+ full papers, and the new method & projection graphs — plus a rigorous public Errata & Evolution record: every correction is documented, nothing is rewritten silently.

100+
Papers
~2200
Pages
2025–2026
Period
📚 View on Zenodo

What's Included

  • Abstract v6.0 (July 2026) — read first: the framework, the honest closure levels, and the human–AI research protocol
  • Canon Live Snapshot — Claim Registry, Conflict Ledger, Open Gaps, Failed Paths, derivation graphs (122 nodes / 177 edges)
  • Method & Projection Graphs — METHOD_GRAPH v0.1.7 (incl. the M-TIME-NONSPATIAL gate), Observable Projection Graph spec + engine
  • Errata & Evolution — the public corrections record: every correction documented
  • Symbol Book v5.1 — Complete parameter registry and derivations
  • Clarification Note — Parameter registry with all 42 definitions
  • Core Theory Papers — 6D geometry, compactification, master equations
  • Gauge Coupling Derivations — Fine structure constant, weak angle, strong coupling
  • Cosmology & DynSys — Dark energy, transition redshift, growth rate
  • N-body Simulations — GADGET-4 with the 3D+3D kernel, up to 768³, physical growth gate passed
  • Observational Tests — SPARC, WALLABY, SLACS, DESI, Planck, PTA
  • Pre-registered Predictions — Euclid, DESI DR2, KATRIN, DUNE constraints

Explainer Video Series: The 3D+3D Framework

Visual Manim animations explaining the theory step-by-step. From the axiom τ = i/φ to 42 parameters and cosmological predictions.

4
Episodes Released
~50 min
Total Duration
▶ Visit Channel
🎬
▶
▶
10:42

Ep 01: The Single Axiom τ = i/φ

Why time has two dimensions and how complex compactification on the golden ratio torus generates all physics.
▶
12:38

Ep 02: Why 6 Dimensions?

The SO(3,3) symmetry structure and how the signature (−,+,+,+,−,−) emerges from topological constraints.
▶
15:17

Ep 03: The Golden Ratio in Physics

How φ ≈ 1.618 appears everywhere: gauge couplings, masses, mixing angles. Unifying principle of all scales.
▶
Coming Soon

Ep 04: Deriving the Electroweak Hierarchy

v = M̄Pl√5·e−32πφ²/7−1/28: the electroweak hierarchy from pure geometry at 0.019%. (Coming Soon)